Lecture
This lecture focuses on solving a Cauchy problem using the general form of constructing a solution for a first-order linear differential equation. The instructor presents the specific equation to be solved, which involves a first derivative and a linear combination of the function and its derivative. The lecture outlines the method for finding the general solution, which includes a constant multiplied by an exponential function and the integral of a given function. The instructor emphasizes the importance of determining the constant that satisfies the initial condition. By substituting the initial value into the general solution, the unique solution to the Cauchy problem is derived. The final solution is presented as a function defined on the interval from zero to infinity, highlighting its global and unique nature. This comprehensive approach provides a clear understanding of the techniques used in solving differential equations and the significance of initial conditions in determining unique solutions.